They won't have the same kinds of resources as a full party (fewer skill points to spend or fewer kinds of spells (only druid, or only wizard, etc)). They'll have trouble since they might have all the druid spells they need, but they won't have all the skills to find and disable that trap, jump the chasm, identify a magic sigil, etc. etc.
They won't have the same number of actions as the enemies they're fighting- four people would get four full rounds of actions and match a party of four they're fighting- two people get half the actions (along with half the hp and such), but are still fighting four full rounds of actions. It's easy for them to get overwhelmed by groups.
Since the player wants to play a druid with an animal companion, you could always say the companion is intelligent and allow the player to give it class levels in place of Animal hit dice (instead of d8 hit die animal hit dice, it takes levels in fighter or something)
This one also helps the player's action economy, as they're running two full-blown characters instead of just one (and a half in the companion). They'll still have fewer actions than the opposing forces most likely, but their actions will be a bit more effective- A fighter 6 lion is much more effective than a plain AC lion with 6 levels of Animal- notably, the fighter one has 2 more BAB and 6 more hp, as well as 4 extra feats and armor proficiency, and is able to take some great fighter archetypes, etc.
There are a ton of ways to balance this one out- have a main character and side characters. You can dole out levels to keep the party as a whole balanced, but it should be okay if the "main" is 2 or so levels higher than the others. (Thanks @LegendaryDude for the additional inspiration)
Consider taking a look at the Black Streams Solo Heroes rules (free) and/or the Scarlet Heroes ruleset (not free), by Kevin Crawford. They adapt D&D to work with a single character, and are designed to handle all the necessary adjustments at a system level, so you don't have to worry about manually adapting each module you encounter. Black Streams is an addon designed to be used with your d&d system of choice; Scarlet Heroes uses its ideas and builds a self-contained ruleset out of it. There is, however, one major issue: they're written for the really old school editions of the game, so some numbers will need adjusting.
First, damage is dramatically reduced. A roll of 1 does no damage, a roll of 2-5 does 1 damage, 6-9 does 2 and more than that does 4. The PC takes this reduced damage to his hp as usual, but enemies effectively have a hp equal to their number of hit dice. So an orc with 2 hit dice who gets hit for 6 damage (pre-reduction) is immediately killed. There are also some additional rules regarding multiple damage dice, and what happens when you take a flat amount of damage rather than rolling. Note that both damage and hp-per-hit-dice are a good deal lower in the old school rules, so these numbers will need adjusting: perhaps halving the PC's hit points (or giving enemies 2hp per HD instead of 1) and then adjusting how much damage gets reduced by according to taste. This also ignores the effects of a high constitution on NPCs and monsters; YMMV on whether you want to come up with a way to adjust for that in some situations.
Healing magic works the same way in reverse; and after a fight you can also take a few moments to catch your breath and restore a few of the HP lost in that fight. This helps if you don't have healing magic of your own, though again the number of HP restored would definitely need to change for Pathfinder. Perhaps 1HP per level might work.
Secondly, if the PC manages to kill an enemy, excess damage can carry over and hit more enemies. Sort of like the 3.5 version of the great cleave feat, except that you don't reroll the attack roll, you just deal the damage left over after your last kill. To avoid exploits, the new target can't have a better AC than your original one.
Thirdly, you're given something called a "fray dice": an automatic dice of damage you can inflict every round, even if you miss with your main attack or are doing something else. This means you're still killing the enemies even if you have a string of bad rolls with all your other attacks, which can happen far more often when you only have one or two a round which hit really hard rather than an entire party's worth that do relatively little damage. There are a few restrictions on what you can hit with it, though: it's for dealing with mooks, not killing bosses. In the original rules, the dice size depends solely on your class; in Pathfinder, though, it should probably scale with your level in some way.
Fourthly, the system provides a mechanic designed to help when they don't have the magic/skill points/etc that a party would normally have access to, or when they're hit by a nasty single-target disable that would normally be dealt with by other party members: defying death. When the PC is in a situation they couldn't otherwise solve (failed a save against a save-or-lose spell, don't have the ability to fly up and pick up a plot coupon, etc) they can invoke this ability to get out of the situation. Perhaps they resist the spell after all, or they spring up to the ceiling in a single mighty bound, snatching the plot coupon as they do so. In return for this, they take a level-based amount of damage, which increases the more times you use the ability in a given day.
Scarlet heroes also switches most d20 rolls (with the notable exception of attack rolls) to 2d8, with target numbers adjusted appropriately. This gives a bell-curve shaped distribution, which helps make things less "swingy": since you'll often find that success or failure depends on a single dice roll, rather than several from different PCs, the normal rules tend to produce rather extreme results more often than you might like.
I've seen a lot of good suggestions that I've used in solo adventures in the past already mentioned. Letting them make a Gestalt character quickly and easily lets them create a stronger character with more skills and abilities, but still leaves them with some exploitable weaknesses.
One of the simplest changes may be editing many of the monsters out of the module and replacing them with simple puzzles or environmental hazards (NOT RIDDLES! What is an obvious answer to one person may not make sense at all to another.) or with social encounters. Although it may be easier to create your own short story line or use the module lore and maps as inspiration for something new. Paizo modules are often VERY combat heavy.
Use the "mook HP" from 4e or other systems, for anything that isn't a mini-boss, special encounter or stronger creature. Give the regular enemies low AC, and a flat amount of HP or 1-3HP per hit die. It will speed up combat and make the player feel like their character really is stronger than their enemies.
Giving them a few more powerful magic items than they would normally have or a minor artifact also helps. Just make sure it's something that is more utilitarian OOC item or a one-shot last resort in combat that they will want to save, rather than use to breeze through everything. I wouldn't give them anything that gives flat bonuses to stats or skills (unless it's a skill they don't have) because that can quickly inflate numbers. Items that give slight damage reduction or elemental resistances can also help keep a single player alive and the story moving even when they would otherwise be down.
Make sure the player has access to potions, scrolls, an item, or person to go to that will heal them, or my favorite, give them a Belt of Healing! They may be a Druid, but if you make them prep half their spells as "Heal X" they won't be thrilled.
Published adventures are generally very easy for a single competently built PC of their expected level. While the loss of resources will increase the challenge of the campaign and fundamentally alter the role-playing opportunities, it does not sufficiently alter the difficulty of a published adventure path to the point that it will be difficult for a T1 class like your player's druid, assuming they build and play their character well.
Pathfinder Adventure Path is a series of monthly publications released by Paizo Inc. Each volume consists of one segment of a multi-part series of adventures linked together by a story arc and theme, with the exception of the special 200th issue, which is a standalone adventure. In addition to the main adventure, each issue also features support articles (2E) on the Pathfinder campaign setting, new monsters, and (through the end of the Strange Aeons Adventure Path) serial Pathfinder's Journal fiction.
Each Adventure Path includes a free Player's Guide that provides a spoiler-free overview of the path, as well as tips and content to help players create characters for the campaign. Some Pathfinder Campaign Setting and Pathfinder Player Companion products are released in concert with specific Adventure Paths to provide related setting context and rules options.
Each Adventure Path is helmed by a lead developer who sees it from concept to outline, and from six raw turnovers from freelance authors into a coherent whole. The first six APs were helmed by Creative Director James Jacobs, who then co-developed Serpent's Skull with Developer Rob McCreary. Beginning with the Carrion Crown Adventure Path, McCreary and Jacobs alternated lead development of APs through Hell's Vengeance. Thereafter, Developers Adam Daigle and Crystal Frasier alternated development on the next four APs. Jacobs returned to the lead developer seat for Return of the Runelords, while Developer Ron Lundeen helmed the Tyrant's Grasp Adventure Path, the last in support of Pathfinder First Edition.
Starting with the Age of Ashes Adventure Path, the line began using Pathfinder Second Edition rules and incorporated the advanced timeline of the Pathfinder Lost Omens product line. Sky King's Tomb was the first Adventure Path to incorporate some of the revisions from the Pathfinder Second Edition Remaster Project, and Wardens of Wildwood is the first to incorporate all of the project's revisions and be licensed under the Open RPG Creative (ORC) license.3
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