Origin of -(e)s in present indicative third singular

I'm aware that it comes from a Northern dialect of Middle English as in:

He sing(e)s

With the full Northern conjugation being:

Ik sing(e)
Þu/ou sing(e)s
He sing(e)s
We/ye/they sings

But in Old English and other West Germanic languages, there wasn't -s or -r in the 3rd-person singular indicative present.

I know that many Northern dialect words come from Old Norse, and a little grammar.

The Norse Conjugation of mæla is:

Ek mæli vér mælum
Þú mælir þér mælið
Hann mælir þeir mæla

And the rune used to write the ending -r in Norse was originally used for the s sound and it remained the rune even when the sound changed, meaning that it was likely originally hann mælis. And there were many Vikings in Northern England.

Is Old Norse a reasonable origin for the third singular -(e)s? Is my little theory sound?

Old English does have the -s ending - just not in West Saxon, the variety that's most commonly described. In the tenth-century Northumbrian Lindisfarne Gospels, for instance, -s is found sometimes in the third person singular as well as in the second person singular and in the plural. The earliest known attestation is in the ninth-century Urswick inscription, which is really too early to show Norse influence. So, without wishing to pour cold water on an intuitively plausible idea, I think it's unlikely that the -s ending itself comes from North Germanic.

(I can provide references for this if you're interested.)

As George points out, this form appeared already in Northumbrian Old English. It appears that the 2nd and 3rd person singular became confused in the northern dialects, perhaps under Scandinavian influence, where the same coalescence took place (and later spread to all other persons and numbers). In his Old English Grammar, A. Campbell says:

A further confusion in North[umbrian] (perhaps agravated by Scandinavian influence) is for 2nd and 3rd sg. to fall together under the 2nd, hence 3rd sg. appears as -es, -as, -aes[1]....

The footnote says:

North[umbrian] has in verbs of all kinds occasionally 2nd sg. in -ð because in 3rd sg. -ð = -s; so Ru[thwell Cross] once hæfeþ habes.

So basically, for some reason, people lost the mental distinction between 2nd and 3rd person singular, and then generally used the 2nd for the 3rd, but sometimes vice versa. Eventually, this spread to the plural, such that all endings but the 1st person singular were "-(e)s".

I can't help but wonder if this was helped by the lack of markings for person in the subjunctive in any dialect of Old English. Also notable is the pre-OE replacement of the 1st and 2nd person plural with the endings of the 3rd person plural. This was so complete that there are absolutely no traces of the original endings anywhere. So there was clearly a trend in OE to mark number and not person so much.