Q:

Overstay in UK 8 years ago - with mention in passport

I over stayed in the UK (Scotland) for more than a year and while returning to India, they wrote (handwritten) 1515/a in my passport. It happened 8 years ago. 
Will there be still any record of that overstay?

A:

they remarked (handwritten) 1515/a in my passport.
The 151a form is the old removal notice, it was replaced in 2014. When they serve you with a removal notice (151/a, 151/b, or the current RED.0001) they can also make an annotation in your passport. The guidance informs us that the overstayer whose breach is more than 90 days must be served, there is no discretionary latitude.  
Unlike refusal notices, removal notices are massively formulaic and there's usually no information of any value to go on for the future. You should hang on to it because it has your case number.  It's the right-most column in the screen shot below...

Will there be still any record with them?
Yes, they keep biometrics for 12 years, but for an overstayer there is no limit.  You need to declare a removal if there is less than 10 years since the date of the incident. If you are considering answering 'no' on the application form, you will get caught.
Note: 2016 minus 8 gives 2008. The government made transitional provisions for overstayers who were caught prior to November 2008 that you may be able to benefit from.  That discussion is off-topic here (IMHO) and best taken up with a solicitor. There are solicitors in London who specialise in word-crafting an explanation that the decision-makers will look favourably on. Alternatively you can ask on the Law site.

A:

I would be very surprised if they don't keep a record of your overstay.
And when you are applying for a visa, I would read very carefully any paper work you were given and think carefully about how you answer the question about deportation on the form

I would tend to feel that, whilst you want to avoid putting 'yes', you also definitely don't want to lie about it.
Either way, you want to look for lots of 'documents' that might be able to support your case that you wont overstay again. The Border force isn't known for its good humour, any excuse to not give a visa is a good excuse for them.

