Thanks all for giving suitable meaning and epithets applicable to Lord Ganesh.
The traditional etymologists offer different ones as I have collected below:
हे = शिवसमीपे, रणे वा, रम्बते इति हेरम्बः;
"हः शंकरे हरौ हंसे रणरोमाञ्चवाजिषु" इति नानार्थरत्नमाला;
हः = Shiva, Hari (Vishnu or Lion), Swan, fighting, erection of hairs (thrilling), horse. Any meaning applicable can be taken in the context. Choice is yours.
It is possible that lion and elephants are mutually enemies as found often described in Sanskrit literature and hence on seeing Ganesha his face being that of an elephant, would produce great sound on seeing Hari (the lion).
One more lexicon for ह -
हः शिवे सलिले शून्ये धारणे मङ्गलेऽपि च।
गगने नकुलीशे च रक्ते नाके च वर्ण्यते॥ इति मेदिनी
Some more addition to the meaning of ह.
हिरते = वर्धयति स्तोतॄन् इति हेरम्बः; "हिर वृद्धौ" + म्ब
हे (इति) रम्बते = शब्दायते वा हेरम्बः।
Some consider this like देशी without any derivation denoting Ganesha (which is rationalistic approach, than inventing any sort of derivation to get the form) = देशीपदप्रायं मन्महे according to some remark in a commentary. क्षीरस्वामी
Really he is a linguist with modern view. He considers it is borrowed from regional languages. According to Mani, there were a tribal people in South India, called हेरम्ब-s inhabitants of हेरम्बक. on the authority of Mahabharata. SabhaParva Ch.31.13.(Gita Press Edn).
प्रत्यूषे हे रम्बते = शब्दायते वा हेरम्बः। इति नैरुक्ता:; देशीपदप्रायं मन्महे।
One episode is alluded to that between the child play, Ganesha squeezing all the heads of Kartikeya with his trunk, hid under the bead-stead hanging on it. Out of severe paid, he shouted, हे तात, , विनायकः, मां प्रहृत्य मञ्चेर(ल)म्बते = लम्बते इति . Thereafter he was called हेरम्ब।
The above is the version of ब्रह्माण्डपुराण.
Free etymology: हेरम्बशब्दं करोतीति वा (हे अम्ब इति शब्दं) करोति; or हे इति रम्बते = शब्दायते - हेरम्बः।
In the opinion of Kshiraswamy, it is loan word from Desi languages (Vernacular) and the same is confirmed by V Mani in his Puranic Encyclopedia referring to a native southern region as .Heramba-hatta referred to in Mahabharata.(was not able to check the edition). Etymological meaning is one thing and the conventional meaning is another thing. Both co-relate sometime in some words. But could not expect them directly follow and attach denotation. It is a commonly accepted view among traditional grammarians as maxim : व्युत्पत्तिनिमित्तमन्यत्, प्रवृत्तिनिमित्तमन्यत्". This view has been examined in Nirukta, while examining the view of 22 जुलाई 2009 –
तत्र नामान्याख्यातजानीति शाकटायनो नैरुक्तसमयश्च । न सर्वाणीति गार्ग्यो, वैयाकरणानाञ्चैके । - निरुक्तम् अ1.12
There is one Herambopanishat too (may be of later origin if common chronology can be applied, but not if Vedic Literature are taken as without origin "anaadi" or "pravaahanitya") which enumerates the popular names along with it.
The Vinayaka- class of retinue of Shiva are referred to in plural number too. The epithet द्वैमातुर denotes जरासन्ध and गणेश also. So Heramba may have more than one referents to signify by etymology according to other norms of mythology rather than etymology.